Relevant conceptual problems: Chapter 2: 1, 2, 3, 5, 6, 7, 8, 10
Problems to be handed in:
1) new This problem gives you practice using the equations given in class over the last week.
You have collected data from an experimental study with two treatments, a control (A) and and a manipulation (B). Sample sizes, averages and standard deviations for each group are:
| Treatment | Sample size | Average | Standard deviation |
| A | 40 | 20 | 8.844 |
| B | 10 | 25 | 10.954 |
a) 1 pt. How many degrees of freedom are associated with the sd for treatment A?
b) 1 pt. Calculate and report the pooled sd
c) 1 pt. How many degrees of freedom are associated with the pooled sd?
d) 1 pt. If you assume that the population variances of the two treatments
are not the same, what is
the standard error of the difference?
e) 1 pt. If you assume that the two population variances are equal, what is
the standard error of the difference?
Note: For all subsequent parts, use the pooled sd, i.e. assume equal variances.
f) 1 pt. Calculate the t statistic that tests the null hypothesis that the difference (as Group B - Group A) = 0. (p-value not needed here or in the next question)
g) 1 pt. Calculate the t statistic that tests the null hypothesis that the difference (as Group B - Group A) = 2.
h) 1 pt. Calculate a 95% confidence interval for the difference in means (as Group B - group A).
i) 1 pt. Calculate a 99% confidence interval for the difference in means (as Group B - group A).
j) 1 pt. Based on the confidence intervals in parts h and i, what can you say about the p-value associated with the t-statistic in part f (null = 0)?
Note: A range of p-values is all I expect.
k) 1 pt. Is it appropriate to conclude that the manipulation (treatment B) has no effect on the response? Briefly explain why or why not.
l) 1 pt. Is it appropriate to conclude that the population means for treatments A and B are equal? Briefly explain why or why not.
2)
Chapter 2: problem 21 Bumpus's sparrows
One of the classic data sets in evolutionary ecology was collected in the late 1890's by Hermon Bumpus at Brown University. After an
especially severe winter storm, he collected and measured moribund sparrows. Some of those sparrows subsequently died
(because of the storm, not the measuring); some did not. Was a sparrows fate (lived, died) associated with its size? The
data I have are the humerus (first wing bone) lengths, in inches, for 35 males that lived and 24 males that died. The data are in
sparrow.csv. The two variables are Humerus (length in inches) and Status (Survived or Perished).
Assume equal variances for parts c, d, and e.
a) 1 pt. Draw side-by-side box plots of humerus lengths in the two groups of sparrows. Your answer is the plot.
b) 1 pt. On average, how much larger are the birds that survived? If the survivors are smaller, report a negative number.
Make sure to include units.
c) 1 pt. What is the pooled standard deviation?
d) 1 pts. What is the standard error of that difference (from part b)? Make sure to include units.
e) 1 pt. Use a t-test to test the null hypothesis of no difference between the two groups. Report the p-value,
f) 1 pt. a 95% confidence interval for the difference,
g) 2 pts. and a short conclusion from your statistical results. I suggest including estimates of the mean humerus lengths or their difference.
General advice:
1) Be sure to check which way around your computer program (SAS, R, JMP)
is computing the difference.
2) You only need to report numbers, e.g. copied from computer output, with units when requested.
3) You do NOT need to hand in copies of computer output. You should keep
a copy of your code so you can compare with our code, in case you get different
results.
4) Remember that the case studies in the Statistical Sleuth provide examples of conclusions. Yours for problem 2g can be much shorter.